diff options
Diffstat (limited to 'FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series')
4 files changed, 243 insertions, 0 deletions
diff --git a/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/README.md b/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/README.md new file mode 100644 index 0000000..1a735f9 --- /dev/null +++ b/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/README.md @@ -0,0 +1,8 @@ +convergence + + +divergence + + +taylor series + diff --git a/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/convergence.py b/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/convergence.py new file mode 100644 index 0000000..fcbcfb4 --- /dev/null +++ b/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/convergence.py @@ -0,0 +1,57 @@ +from manimlib.imports import * +def GetCenters(width,center,n): + d = width / 4 + list = [center + [0,d,0]] + if n > 1: + list.append(center + [-d,-d,0]) + if n > 2: + list.extend(GetCenters(width / 2, center + [d,-d,0],n-2)) + return list +END_CENTERS = [ORIGIN] +END_CENTERS.extend(GetCenters(3, 3 * RIGHT, 24)) +color_list = ['#00931F','#A93226','#D68910','#17A589','#2471A3','#884EA0','#E74C3C','#D4AC0D'] +COLORS = [color_list[i % len(color_list)] for i in range(50)] +class RectangleFromSequence(Rectangle): + CONFIG = { + "sequence_number": 0, + "center": ORIGIN + } + def __init__(self, **kwargs): + digest_config(self, kwargs) + Rectangle.__init__(self,height = 3 * (1/2) ** ((self.sequence_number + 1) // 2),width = 3 * (1/2) ** ((self.sequence_number) // 2),**kwargs) + if self.sequence_number < 6: + if self.sequence_number == 0: + label = TexMobject("1") + else: + label = TexMobject("1/",str(2 ** self.sequence_number)) + label.scale(0.8 ** self.sequence_number) + self.add(label) + self.label = label + self.set_fill(COLORS[self.sequence_number],1) + self.set_stroke(width = 1) + self.move_to(self.center) +equation = TexMobject("\\sum_{n=0}^\\infty \\frac{1}{2^n} =","1","+","\\frac{1}{2}","+","\\frac{1}{4}","+","\\frac{1}{8}","+","\\frac{1}{16}","+ \\ldots","= 2") +class Proof1(Scene): + def construct(self): + equation.to_edge(UL) + self.play(Write(equation[0:-1])) + rects = VGroup(*[RectangleFromSequence(sequence_number = i)for i in range(25)]) + rects.arrange(RIGHT, buff=0.5) + left_center = 5*LEFT + rects.shift(left_center-rects[0].get_center()) + for rect in rects: + rect.shift(DOWN*rect.get_top()+UP*3 / 2) + for i in range(25): + rects[i].generate_target() + rects[i].target.move_to(left_center+END_CENTERS[i]) + self.wait() + for i in range(5): + self.play(GrowFromPoint(rects[i] , equation[2*i+1].get_center())) + self.play(*[GrowFromPoint(rects[i] , equation[-2].get_center())for i in range(5,25)]) + self.wait() + for i in range(1,8): + self.play(MoveToTarget(rects[i])) + self.play(*[MoveToTarget(rects[i]) for i in range(8,25)]) + self.wait(0.5) + self.play(Write(equation[-1])) + self.wait(3)
\ No newline at end of file diff --git a/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/divergence.py b/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/divergence.py new file mode 100644 index 0000000..4f4bf7c --- /dev/null +++ b/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/divergence.py @@ -0,0 +1,111 @@ +from manimlib.imports import * +class divergence(GraphScene): + CONFIG = { + "y_max" : 2, + "y_min" : -2, + "x_max" : 20, + "x_min" : 0, + "y_tick_frequency" : 1, + "x_tick_frequency" : 1, + "axes_color" : WHITE, + "num_graph_anchor_points": 3000, + "graph_origin" : ORIGIN+6*LEFT, + "x_labeled_nums": None, + "y_labeled_nums": [-2,-1,1,2], + "x_axis_label":r"${(-1)}^{n}$", + "y_axis_label":"$Sum$", + "x_axis_width": 10, + "y_axis_height": 4, + } + def construct(self): + XTD = self.x_axis_width/(self.x_max - self.x_min) + YTD = self.y_axis_height/(self.y_max - self.y_min) + text1 = TextMobject("Consider the series 1-1+1-1+1-1+1-......") + self.add(text1) + self.wait(3) + self.play(FadeOut(text1)) + self.setup_axes() + rangeo = (20)//self.x_axis_width + for i in range(0,2): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+0.5*RIGHT) + self.add(texta) + for i in range(2,4): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+0.65*RIGHT) + self.add(texta) + for i in range(4,6): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+0.8*RIGHT) + self.add(texta) + for i in range(6,8): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+0.95*RIGHT) + self.add(texta) + for i in range(8,10): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+1.1*RIGHT) + self.add(texta) + for i in range(10,12): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+1.35*RIGHT) + self.add(texta) + for i in range(12,14): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+1.5*RIGHT) + self.add(texta) + for i in range(14,16): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+1.65*RIGHT) + self.add(texta) + for i in range(16,18): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+1.8*RIGHT) + self.add(texta) + for i in range(18,20): + texta = TextMobject(str((-1)**i)).move_to(self.graph_origin+0.2*(rangeo*i)*RIGHT+0.5*DOWN+1.95*RIGHT) + self.add(texta) + + text2 = TextMobject("Number of purple lines denotes the number of terms added").move_to(1*UP).scale(0.8) + self.play(ShowCreation(text2)) + self.wait(4) + self.play(FadeOut(text2)) + for i in range(0,2): + horline = Line(np.array([-5.5+i,1,0]), np.array([-5+i,1,0]), color = PINK) + verline = DashedVMobject(Line(np.array([-5+i,1,0]), np.array([-5+i,0,0]))) + botline = Line(np.array([-6+i,0,0]), np.array([-5.5+i,0,0]), color = PINK) + upline = DashedVMobject(Line(np.array([-5.5+i,0,0]), np.array([-5.5+i,1,0]))) + self.play(ShowCreation(botline), run_time = 0.2) + self.play(ShowCreation(upline), run_time = 0.2) + self.play(ShowCreation(horline), run_time = 0.2) + self.play(ShowCreation(verline), run_time = 0.2) + + text3 = TextMobject("For even number of terms, sum is 0").move_to(1*UP).scale(0.8) + self.play(FadeIn(text3)) + self.wait(4) + self.play(FadeOut(text3)) + for i in range(2,4): + horline = Line(np.array([-5.5+i,1,0]), np.array([-5+i,1,0]), color = PINK) + verline = DashedVMobject(Line(np.array([-5+i,1,0]), np.array([-5+i,0,0]))) + botline = Line(np.array([-6+i,0,0]), np.array([-5.5+i,0,0]), color = PINK) + upline = DashedVMobject(Line(np.array([-5.5+i,0,0]), np.array([-5.5+i,1,0]))) + self.play(ShowCreation(botline), run_time = 0.2) + self.play(ShowCreation(upline), run_time = 0.2) + self.play(ShowCreation(horline), run_time = 0.2) + self.play(ShowCreation(verline), run_time = 0.2) + botline = Line(np.array([-6+4,0,0]), np.array([-5.5+4,0,0]), color = PINK) + upline = DashedVMobject(Line(np.array([-5.5+4,0,0]), np.array([-5.5+4,1,0]))) + self.play(ShowCreation(botline)) + self.play(ShowCreation(upline)) + text4 = TextMobject("For odd number of terms, sum is 1").move_to(1.5*UP).scale(0.8) + self.play(FadeIn(text4)) + self.wait(3) + for i in range(4,10): + horline = Line(np.array([-5.5+i,1,0]), np.array([-5+i,1,0]), color = PINK) + verline = DashedVMobject(Line(np.array([-5+i,1,0]), np.array([-5+i,0,0]))) + botline = Line(np.array([-6+i,0,0]), np.array([-5.5+i,0,0]), color = PINK) + upline = DashedVMobject(Line(np.array([-5.5+i,0,0]), np.array([-5.5+i,1,0]))) + self.play(ShowCreation(botline), run_time = 0.2) + self.play(ShowCreation(upline), run_time = 0.2) + self.play(ShowCreation(horline), run_time = 0.2) + self.play(ShowCreation(verline), run_time = 0.2) + text5 = TextMobject("The sum is oscillating between 1 and 0").move_to(1.5*UP).scale(0.8) + self.play(ReplacementTransform(text4, text5)) + self.wait(4) + text6 = TextMobject("It does not coerce to a particular finite value").move_to(1.5*UP).scale(0.8) + self.play(ReplacementTransform(text5, text6)) + self.wait(4) + text7 = TextMobject("Hence it diverges").move_to(1.5*UP).scale(0.8) + self.play(ReplacementTransform(text6, text7)) + self.wait(3)
\ No newline at end of file diff --git a/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/taylorseries.py b/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/taylorseries.py new file mode 100644 index 0000000..2e9d423 --- /dev/null +++ b/FSF-2020/calculus/intro-to-calculus/infinite-seq-and-series/taylorseries.py @@ -0,0 +1,67 @@ +from manimlib.imports import * +class conv(GraphScene): + CONFIG = { + "y_max" : 10, + "y_min" : 0, + "x_max" : 5, + "x_min" : -1, + "y_tick_frequency" : 5, + "x_tick_frequency" : 1, + "axes_color" : WHITE, + "num_graph_anchor_points": 3000, + "graph_origin" : ORIGIN+2*DOWN+6*LEFT, + "x_labeled_nums": list(range(0,6)), + "y_labeled_nums": list(range(0,11))[::1], + "x_axis_label":"x", + "y_axis_label":"$f(x)$", + "x_axis_width": 9, + "y_axis_height": 5, + } + def construct(self): + XTD = self.x_axis_width/(self.x_max - self.x_min) + YTD = self.y_axis_height/(self.y_max - self.y_min) + texta = TexMobject(r"\text{Expressing }", r"{e}^{x}", r"\text{ as its Taylor series}") + self.play(FadeIn(texta)) + self.wait(3) + self.play(FadeOut(texta)) + self.setup_axes() + graph1 = self.get_graph(lambda x : (np.e)**x, x_min = -1, x_max = 5, color = PINK) + graph2 = self.get_graph(lambda x : 1+x, x_min = -1, x_max = 5, color = GREEN_SCREEN) + graph3 = self.get_graph(lambda x : 1+(x)+(x**2)/2, x_min = -1, x_max = 5, color = GREEN_SCREEN) + graph4 = self.get_graph(lambda x : 1+(x)+(x**2)/2+(x**3)/6, x_min = -1, x_max = 5, color = GREEN_SCREEN) + graph5 = self.get_graph(lambda x : 1+(x)+(x**2)/2+(x**3)/6+(x**4)/24, x_min = -1, x_max = 5, color = GREEN_SCREEN) + graph6 = self.get_graph(lambda x : 1+(x)+(x**2)/2+(x**3)/6+(x**4)/24+(x**5)/120, x_min = -1, x_max = 5, color = GREEN_SCREEN) + graph7 = self.get_graph(lambda x : 1+(x)+(x**2)/2+(x**3)/6+(x**4)/24+(x**5)/120+(x**6)/720, x_min = -1, x_max = 5, color = GREEN_SCREEN) + graph8 = self.get_graph(lambda x : 1+(x)+(x**2)/2+(x**3)/6+(x**4)/24+(x**5)/120+(x**6)/720+(x**7)/5040, x_min = -1, x_max = 5, color = GREEN_SCREEN) + texta = TexMobject(r"{e}^{x}").move_to(self.graph_origin+ 1*RIGHT+2*UP) + textb = TexMobject(r"\therefore {e}^{x}=").move_to(3.5*LEFT+0.5*DOWN) + text1 = TexMobject(r"1+x").move_to(4*RIGHT) + text2 = TexMobject(r"+\frac{{x}^{2}}{2!}").move_to(4*RIGHT) + text3 = TexMobject(r"+\frac{{x}^{3}}{3!}").move_to(4*RIGHT) + text4 = TexMobject(r"+\frac{{x}^{4}}{4!}").move_to(4*RIGHT) + text5 = TexMobject(r"+\frac{{x}^{5}}{5!}").move_to(4*RIGHT) + text6 = TexMobject(r"+\frac{{x}^{6}}{6!}").move_to(4*RIGHT) + text7 = TexMobject(r"+\frac{{x}^{7}}{7!}+...").move_to(4.5*RIGHT) + + self.play(ShowCreation(texta)) + self.play(ShowCreation(graph1)) + self.wait(3) + self.play(FadeOut(texta)) + self.play(ShowCreation(graph2)) + self.play(ShowCreation(text1)) + self.wait(3) + self.play(ReplacementTransform(graph2, graph3), ApplyMethod(text1.shift, 1*LEFT), ShowCreation(text2)) + self.wait(3) + self.play(ReplacementTransform(graph3, graph4), ApplyMethod(text1.shift, 1*LEFT), ApplyMethod(text2.shift, 1*LEFT), ShowCreation(text3)) + self.wait(3) + self.play(ReplacementTransform(graph4, graph5), ApplyMethod(text1.shift, 1*LEFT), ApplyMethod(text2.shift, 1*LEFT), ApplyMethod(text3.shift, 1*LEFT), ShowCreation(text4)) + self.wait(3) + self.play(ReplacementTransform(graph5, graph6), ApplyMethod(text1.shift, 1*LEFT), ApplyMethod(text2.shift, 1*LEFT), ApplyMethod(text3.shift, 1*LEFT), ApplyMethod(text4.shift, 1*LEFT), ShowCreation(text5)) + self.wait(3) + self.play(ReplacementTransform(graph6, graph7), ApplyMethod(text1.shift, 1*LEFT), ApplyMethod(text2.shift, 1*LEFT), ApplyMethod(text3.shift, 1*LEFT), ApplyMethod(text4.shift, 1*LEFT), ApplyMethod(text5.shift, 1*LEFT), ShowCreation(text6)) + self.wait(3) + self.play(ReplacementTransform(graph7, graph8),ApplyMethod(text1.shift, 1*LEFT), ApplyMethod(text2.shift, 1*LEFT), ApplyMethod(text3.shift, 1*LEFT), ApplyMethod(text4.shift, 1*LEFT), ApplyMethod(text5.shift, 1*LEFT), ApplyMethod(text6.shift, 1*LEFT), ShowCreation(text7)) + self.wait(3) + grp = VGroup(text1, text2, text3, text4, text5, text6, text7) + self.play(ApplyMethod(grp.shift, 0.5*DOWN), FadeIn(textb)) + self.wait(5)
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